## OGQR 2020: Question No. 94

In the racetrack shown above, regions I and III are semi-circular with radius r. If region II is rectangular and its length is twice its width, what is the perimeter of the track in terms of r?

Source | OGQR 2020 |

Type | Problem Solving |

Topic | Geometry |

Sub-Topic | Circle / Rectangle |

Difficulty | Medium |

### Solution

__Given__

__Given__

In this question, we are given

- The diagram of a racetrack consists of two semi-circular regions, with radius r, and one rectangular region.
- The length of the rectangular region is twice its width.

__To Find__

__To Find__

We need to determine

- The perimeter of the track, expressed in terms of r.

__Approach & Working__

__Approach & Working__

The perimeter of each semi-circular region = πr

- Therefore, the perimeter of both semi-circular regions together = 2πr

The rectangular region has length twice its width.

- Width of the rectangular region = diameter of the semi-circular region = 2r
- Hence, length of the rectangular region = 2 * 2r = 4r

Therefore, the perimeter of the whole region = 2πr + 4r + 4r = 2πr + 8r = 2r (π + 4)

Hence, the correct answer is option B.

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